Mathematical structure of the Bethe-Salpeter equation for massless exchange reinvestigated

Solutions of the Bethe-Salpeter (BS) equation for two scalar particles of arbitrary masses interacting through an exchange of a massless scalar particle are reinvestigated by means of a new method, applicable to the problem irrespective of the metric (Euclidean or pseudo-Euclidean) of the underlying four-momentum space. This enables one to perform a step-by-step comparison of the solving procedure as applied on one hand to the "original" equation (with underlying pseudo-Euclidean metric), and to the corresponding Wick equation (obtained from the former by formally performing the Wick rotation without prior proof of its actual validity) on the other hand. At a certain point (compare the results of Secs. IV and V, respectively) the kernels of the appropriate transformed integral equations corresponding to the two cases become manifestly analytically different. This finding seems not only to render the Wick rotation—$a$ posteriori—invalid, but also to preclude one—in the realistic (i.e., "original") case—from obtaining the well-known Wick-Cutkosky solutions (reproduced fully in the case of the Wick equation). Although the "original" version of the BS equation is thus found too difficult to solve exactly (due to the presence of an additional parameter in the kernel), the method developed leads in a most natural way to an exactly soluble model of the BS equation obtained by retaining the pseudo-Euclidean metric but replacing the Feynman propagator $D_F$ for the exchange particle by the "relativistic Coulomb" propagator $\overline{D}$ (half the difference between "advanced" and "retarded" propagators). This model exhibits a marked correspondence—in its nonrelativistic limit—with the Schrödinger solution of the Coulomb problem. It should finally be noted that our method avoids any series expansions of the results what-soever (partial-wave expansion included) which would otherwise tend to obscure clear-cut "analytic" conclusions by posing convergence problems, and thus aims always at obtaining a closed-form expression for the total (off mass shell) scattering amplitude.